The Strassen invariance principle for certain non-stationary Markov–Feller chains

Article type: Research Article

Authors: Czapla, Dawid | Horbacz, Katarzyna | Wojewódka-Ściążko, Hanna^{; *}

Affiliations: Institute of Mathematics, University of Silesia in Katowice, Bankowa 14, 40-007 Katowice, Poland. E-mails: [email protected], [email protected], [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We propose certain conditions implying the functional law of the iterated logarithm (the Strassen invariance principle) for some general class of non-stationary Markov–Feller chains. This class may be briefly specified by the following two properties: firstly, the transition operator of the chain under consideration enjoys a non-linear Lyapunov-type condition, and secondly, there exists an appropriate Markovian coupling whose transition probability function can be decomposed into two parts, one of which is contractive and dominant in some sense. Our criterion may serve as a useful tool in verifying the functional law of the iterated logarithm for certain random dynamical systems, developed e.g. in biology and population dynamics. In the final part of the paper we present an example application of our main theorem to a mathematical model describing stochastic dynamics of gene expression.

Keywords: Markov chain, random dynamical system, invariant measure, law of the iterated logarithm, asymptotic coupling

DOI: 10.3233/ASY-191592

Journal: Asymptotic Analysis, vol. 121, no. 1, pp. 1-34, 2021